Problem 29 - Algebra 2

Question

Geometry The sides of a triangle are 15 in., 17 in., and 16 in. long. The smallest angle has a measure of \(54^{\circ} .\) Find the measure of the largest angle. Round your answer to the nearest degree.

Step-by-Step Solution

Answer

The measure of the largest angle in the triangle is \(66^{\circ}\).

Step 1: Identify the given

It's given that the sides of the triangle are 15 in., 16 in., and 17 in., while the smallest angle is \(54^{\circ}\). Now, note that the largest angle will be facing the longest side, which is 17 in. in this case.

Step 2: Apply the Law of Cosines

The Law of Cosines is: \(c^{2} = a^{2} + b^{2} - 2ab \cdot cos(C)\), where \(a\), \(b\) and \(c\) are the side lengths, and \(C\) is the angle opposite side \(c\). In this case, \(a = 15\) in., \(b = 16\) in., \(c = 17\) in., and \(C\) is the largest angle we're trying to find.

Step 3: Use the formula to find the unknown angle

Substitute 15 for \(a\), 16 for \(b\), and 17 for \(c\) into our equation to solve for \(cos(C)\) first. We have \(289 = 225 + 256 - 2 \cdot 15 \cdot 16 \cdot cos(C)\), which solves to \(cos(C) = 0.4045\). Now, use the arccos function to find the measure of the angle \(C\), which is \(arcos(0.4045) = 66.4^{\circ}\). Round this to the nearest degree to get \(66^{\circ}\).

Key Concepts

Triangle Geometry

Triangles are fundamental shapes in geometry, consisting of three sides, three angles, and three vertices. Let's break down the triangle from the exercise, where the side lengths are 15 in., 16 in., and 17 in.:

• A triangle with all sides of different lengths is called a scalene triangle. All angles in a scalene triangle are also different.
• The angle opposite to the longest side is always the largest angle in such triangles.
• The sum of all interior angles in any triangle is always equal to 180 degrees.

Understanding these basic properties helps in applying trigonometric rules like the Law of Cosines effectively.

Largest Angle Calculation

Calculating the largest angle in a triangle can be efficiently done using the Law of Cosines. This rule relates the lengths of a triangle’s sides to the cosine of one of its angles.
The formula is given as:\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]In this equation:

• The sides labeled as \(a\), \(b\), and \(c\) represent the side lengths of the triangle.
• \(C\) represents the angle opposite side \(c\).

In the given exercise, the longest side of 17 inches is opposite the largest angle (\(C\)). By plugging the side lengths into the formula and solving for \(\cos(C)\), we find the largest angle by taking the arccosine. This gives us an angle of approximately \(66^{\circ}\) after rounding.

Trigonometry Concepts

Trigonometry helps us understand the relationships between the angles and sides of triangles. Key concepts used in this exercise include:

• Law of Cosines: This is used when you know all three sides of the triangle and need to find an unknown angle or vice versa.
• Cosine Function: A trigonometric function that relates the angle to the adjacent side over the hypotenuse in a right triangle. For non-right triangles like our scalene triangle, it's used in the Law of Cosines.
• Arccos Function: This is the inverse of the cosine function, allowing us to calculate angles from cosine values.

By combining these trigonometric concepts, we can accurately calculate unknown angles and sides in complex triangles, making them easier to analyze.